The rate of spatial decay of nonnegative solutions of nonlinear parabolic equations and inequalities

Abstract

Let L L be a uniformly parabolic linear partial differential operator. We show that nonnegative solutions of the differential inequality L u ≤ c ( u + | ∇ u | ) Lu \leq c(u + |\nabla u|) on R n × ( 0 , T ) {{\mathbf {R}}^n} \times (0,T) for which u ( x , T ) = 0 ( exp ⁡ ( − δ | x | 2 ) ) u(x,T) = {\mathbf {0}}(\exp {\text {(}} - \delta |x{|^2})) must be identically zero if the constant δ \delta is sufficiently large. An analogous result is given for nonlinear systems.